Spectral sets and factorizations of finite abelian groups (Q1354605)

A spectral set is a subset \(\Omega\) of \(\mathbb{R}^n\) with Lebesgue measure \(0<\mu(\Omega)< \infty\) such that there exists a set \(\Lambda\) of exponential functions which form an orthogonal basis of \(L^2(\Omega)\). The spectral set conjecture of B. Fuglede states that a set \(\Omega\) is a spectral set if and only if \(\Omega\) tiles \(\mathbb{R}^n\) by translation. The authors studied sets \(\Omega\) which tile \(\mathbb{R}^n\) using a rational periodic tile set \(\varphi= \mathbb{Z}''+ A\), where \(A\subseteq(1/N_1)\mathbb{Z}\times \cdots\times (1/N_n)\mathbb{Z}\) is finite. They characterized geometrically bounded measurable sets \(\Omega\) that tile \(\mathbb{R}^n\) with such a tile set. Certain tile sets \(\varphi\) have the property that every bounded measurable set \(\Omega\) which tiles \(\mathbb{R}^n\) with \(\varphi\) is a spectral set, with a fixed spectrum \(A_\varphi\). The authors call \(\Lambda_\varphi\) a univeral spectrum for such \(\varphi\). They gave a necessary and sufficient condition for a rational periodic set \(\Lambda\) to be a universal spectrum for \(\varphi\), which is expressed in terms of factorization \(A\oplus B= G\), where \(G= Z_{N_1}\times\cdots \times Z_{N_n}\) and \(A:= A(\text{mod }\mathbb{Z}^n)\). In dimension \(n=1\), the authors showed that \(\varphi\) has a universal spectrum whenever \(N_1\) is the order of ``good'' group in the sense of Hajos, and for various other sets \(\varphi\).